Given : the plane region R is bounded by the curves ? = 0 , ? = 2 , ? = ³√?

2-Compute the volume of the solid that has the region R as the (horizontal) base; vertical slices parallel to the ?-axis are squares

use the a) midpoint rule, b) Trapezoidal rule, and c) the Simpsons rule to approximate the given integral with the value of n and round to 4 decimal places

integral (from 0 to 1) e^-x^2 dx, n = 10

show work please

a) State the definition that a function f(x) is continuous at x = a. b) Let f(x) = ax^2 + b if 0 < x ≤ 2

18/x+1 if x > 2

If f(1) = 3, determine the values of a and b for which f(x) is continuous for all x > 0.

2.) g(x) = 4√x − x²

2f.) Label each critical value as a local maximum or minimum. Show Work.

Local Maximum(s) at x = ____________ Local Minimum(s) at x = ____________

2g.) Determine the intervals in which y=g(x) is increasing/decreasing

g(x) increasing: ____________ g(x) decreasing: _

2h.) Determine the intervals in which y=f(x) is concave up/concave down.

g(x) concave up: ____________ g(x) concave down: ____________

2i.) Find the point(s) of inflection

Point(s) of Inflection at x = ______

2j.) lim x→∞ g(x) = _______

2k.) lim x→−∞ g(x) = _______

2l.) Use everything you determined in #2a-k to draw a nice graph. (Draw your own axes)

5.22.

Determine the standard form of the following conics:
(a) 13x^2 − 10xy + 13y^2 − 12√2x + 60√2y + 72 = 0.
(b) 6x^2 + 12xy + 6y^2 − 35√2x − 37√2y + 118 = 0.
(c) 11x^2 − 6x√3y − 6x√3 + y^2 + 2y − 63 = 0

An equation of a hyperbola is given.

y^2/36 - x^2/64 = 1

(a) Find the vertices, foci, and asymptotes of the hyperbola. (Enter your asymptotes as a comma-separated list of equations.)

(b) Determine the length of the transverse axis.

(c) Sketch a graph of the hyperbola.

A pig farmer wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle (see the figure below). There are 900 feet of fencing available to complete the job. What is the largest possible total area of the four pens?

The answer is not 40500 ft^2.

Find the center of mass of a thin plate of constant density delta covering the given region. The region bounded by the parabola x= 6y^2 -3y and the line x= 3y. Please post all steps.

A Toyota Prius is a full hybrid electric automobile that costs $30,000 with 55 mpg. A Toyota Yaris is a normal car that runs on gasoline, and costs $15,000 with 35 mpg. After 100,000 miles, the Prius requires a battery replacement that costs $3,000. Suppose you drive 10 miles per day and 1 gallon of gas costs $4. How much would you have to spend on gas for each car after 10 years? Which car would cost you more money after 10 years? How much different are the gas prices between the Prius and the Yaris (10 miles per day x 10 years)?

1. A 100-gallon tank initially holds 50 gal of brine containing 5 pounds of salt. Brine
containing 1 pound of salt per gallon enters the tank at the rate of 4 gal/s, and the
well-mixed brine in the tank flows out at the rate of 3 gal/s.
(a) How much salt will the tank contain at the time when the tank is full of brine?
(b) If the tank had an infinite capacity, what is the maximum amount of salt that
it could contain?

4. y' − 2xy = 8x. find a general solution.

Find the eigenvalues

λ_n

and eigenfunctions

y_n(x)

for the given boundary-value problem. (Give your answers in terms of n, making sure that each value of n corresponds to a unique eigenvalue.)

y'' + λy = 0,  y(0) = 0,  y(π/6) = 0

Explain in your own words, what can happen if D = 0 (Hessian Determinant).
Given in particular an example where we have a maximum, where we have
a minimum, or where we have neither a maximum nor a minimum.

A shelf in the Metro Department Store contains 95 colored ink cartridges for a popular ink-jet printer. six of the cartridges are defective. (a) If a customer selects 2 cartridges at random from the shelf, what is the probability that they are all defective? (Round your answer to five decimal places.) (b) If a customer selects 2 cartridges at random from the shelf, what is the probability that at least 1 is defective? (Round your answer to three decimal places.)

Find the absolute maximum and minimum values of f on the set D. Also note the point(s) where these absolute maximum and minimum values are located. f(x, y) = 9x^2 + 36x^2 y - 4y - 1 D is the region described as follows: D = { (x,y) | -2 < x < 3; -1 < y < 4}